A cascade
C
C
is defined as a sum of binomial coefficients
\[
C
=
(
a
h
h
)
+
(
a
h
−
1
h
−
1
)
+
⋯
+
(
a
t
t
)
C = \left ( {\begin {array}{*{20}{c}} {{a_h}} \\ h \\ \end {array} } \right ) + \left ( {\begin {array}{*{20}{c}} {{a_{h - 1}}} \\ {h - 1} \\ \end {array} } \right ) + \cdots + \left ( {\begin {array}{*{20}{c}} {{a_t}} \\ t \\ \end {array} } \right )
\]
where
a
h
>
a
h
−
1
>
⋯
>
a
t
{a_h} > {a_{h - 1}} > \cdots > {a_t}
. In this expression, we assume that
(
h
a
)
=
0
(_h^a) = 0
whenever
a
>
h
a > h
. Given a cascade
C
C
and a sequence
ε
=
⟨
ε
h
,
ε
h
−
1
,
…
,
ε
t
⟩
\varepsilon = \langle {\varepsilon _h},{\varepsilon _{h - 1}}, \ldots ,{\varepsilon _t}\rangle
of signs (i.e.
ε
i
=
+
1
or
−
1
{\varepsilon _i} = + 1\;{\text {or}}\; - 1
for each
i
i
), we define
\[
ε
C
=
ε
h
(
a
h
h
)
+
⋯
+
ε
t
(
a
t
t
)
.
\varepsilon C = {\varepsilon _h}\left ( {\begin {array}{*{20}{c}} {{a_h}} \\ h \\ \end {array} } \right ) + \cdots + {\varepsilon _t}\left ( {\begin {array}{*{20}{c}} {{a_t}} \\ t \\ \end {array} } \right ).
\]
Also, we put
\[
α
C
=
(
a
h
h
+
1
)
+
(
a
h
−
1
h
)
+
⋯
+
(
a
t
t
+
1
)
.
\alpha C = \left ( {\begin {array}{*{20}{c}} {{a_h}} \\ {h + 1} \\ \end {array} } \right ) + \left ( {\begin {array}{*{20}{c}} {{a_{h - 1}}} \\ h \\ \end {array} } \right ) + \cdots + \left ( {\begin {array}{*{20}{c}} {{a_t}} \\ {t + 1} \\ \end {array} } \right ).
\]
In the paper, we prove that for any sequence
⟨
n
0
,
n
1
,
…
,
n
s
⟩
\langle {n_0},\,{n_1}, \ldots ,{n_s}\rangle
of integers, there exist a cascade
C
C
and a corresponding sequence
ε
\varepsilon
of signs such that
n
i
=
ε
α
i
C
{n_i} = \varepsilon {\alpha ^i}C
for
i
=
0
,
1
,
…
,
s
i = 0,\;1, \ldots ,s
where
α
0
C
=
C
,
α
1
C
=
α
C
,
α
2
C
=
α
(
α
1
C
)
{\alpha ^0}C = C,\;{\alpha ^1}C = \alpha C,\;{\alpha ^2}C = \alpha ({\alpha ^1}C)
, and recursively,
α
n
C
=
α
(
α
n
−
1
C
)
{\alpha ^n}C = \alpha ({\alpha ^{n - 1}}C)
.